On finite state automaton actions of HNN extensions of free abelian groups

HNN extensions of free abelian groups are considered. For arbitrary prime $p$ it is introduced a class of such extensions that act by finite automaton permutations over an alphabet $ \mathsf{X} $ of cardinality $p$ and belong to $p$-Sylow subgroup of the group of automaton permutations over such $ \mathsf{X} $. As a corollary it implies that all corresponding HNN extensions are residually $p$-finite.